As long as you have paperwork proving they owe you, and you have been through all the effort of trying to get them to pay, the first thing you must do is send them a letter warning them that they have 7 days from it's receipt to pay up, or face a county court claim. Send the letter by registered post so you can track when it was received.
If you don't get a reply by the end of business on the last day of the seven days, DO NOT waste any time, file a claim immediately with Money Claim Online, claiming the outstanding money, the court fees, reasonable costs for yourself (postage, phone calls, time spent on it times the minimum wage rate, etc, etc), plus the all important 8% compound interest on the outstanding amount from the day it was originally owed. There are calculators online that will work out the amount of interest for you, and the daily rate it is increasing by.
They then have 14 days I believe (though it may have changed) to either admit the claim or contest it, from the sound of it they have no contest , when they admit it, they will have to cough up, if they fail to answer the court papers, you will win by default.
The court will set a date by which they have to pay the money, if they fail, you can start the procedure to send in the bailiffs, most defaulting companies will pay up when they see this happening, but it will usually happen before.
Unfortunately you will have to pay the fees up front, but these are 100% reclaimable as part of the court action.